Mathematics - Straight Lines Question with Solution | TestHub

$x+y=1$ meets $x$-axis at $A(1,0)$ and $y$-axis at $B(0,1)$.

The coordinates of P are $(1/2,1/2)$ and $P{P}_1$ is perpendicular to OA .

$\Rightarrow O{P}_1=P_{1}P=1/2$

Equation of line OP is $y=x$

We have ${\left(O{M}_{n-1}\right)}^2={\left(O{P}_n\right)}^2+{\left(P_n{M}_{n-1}\right)}^2=2{\left(O{P}_n\right)}^2=2P_n^2$ (say) Also, ${\left(O{P}_{n-1}\right)}^2={\left(O{M}_{n-1}\right)}^2+{\left(P_{n-1}{M}_{n-1}\right)}^2$ ${\left(O{P}_{n-1}\right)}^2={\left(O{M}_{n-1}\right)}^2+{\left(P_{n-1}{M}_{n-1}\right)}^2=2p_n^2+\frac{1}{2}{p}_{n-1}^2$ $\Rightarrow p_n^2=\frac{1}{4}{p}_{n-1}^2\Rightarrow p_n=\frac{1}{2}{p}_{n-1}$ $\therefore O{P}_n=p_n=\frac{1}{2}{p}_{n-1}=\frac{1}{2^2}{p}_{n-2}=\dots \dots \dots =\frac{1}{2^{n-1}}{p}_1=\frac{1}{2^n}$